Multifrequency electrical impedance imaging using lorentz fields

ABSTRACT

In this invention, an approach is proposed to image the electrical impedance (conductivity and permittivity) properties of biological tissues. This approach is based on electrical current induction using ultrasound together with an applied static magnetic field. Acoustic vibrations are generated via piezoelectric transducers located on the surface of a biological body. A linear phased array piezoelectric transducer is used to form pressure distribution in human body/tissue. In the existence of a static magnetic field, the resultant (velocity) current density is sensed by a receiver coil encircling the tissue or placed near the tissue and used for reconstructing the conductivity and permittivity distribution.

FIELD OF THE INVENTION

The present invention is related to medical imaging, in particular to electrical impedance imaging.

STATE OF THE ART

Imaging electrical conductivity of biological tissues is one of the major research areas in the field of medical imaging. To image the conductivity of body tissues researchers have proposed different approaches. The earliest and generally accepted title for this relatively new imaging modality is Electrical Impedance Tomography (EIT) which uses surface electrodes to inject current and measures voltages from the body surface. Magnetic Resonance Electrical Impedance Imaging Tomography (MREIT), Magnetic Induction Tomography (MIT), Magneto Acoustic Tomography (MAT), Magneto-Acousto-Electrical Tomography (MAET) and Magneto Acoustic Tomography with Magnetic Induction (MAT-MI) are approaches proposed for the same purpose, but these methods use different means for applying currents and measurements.

In EIT, there are two major approaches: Applied Current Electrical Impedance Tomography (ACEIT) [1-3] and Induced Current Electrical Impedance Tomography (ICEIT) [4, 5]. In ACEIT electrodes are placed on the surface of a body. Electrical current is injected between a pair of electrodes and the resultant voltage is recorded between two surface electrodes [1]. Current drive and voltage measurement electrodes are changed to obtain an independent set of data for image reconstruction.

In MIT [6-9], currents are induced in the conductive body by time-varying magnetic fields using transmitter coils. The secondary magnetic fields due to existence of the conductive body are measured using receiver coils. The number of independent measurements is increased by changing the location of the transmitter coil.

In addition to these early attempts performed electromagnetically, recently, novel techniques are proposed that integrates electromagnetism with acoustics. MAT [10-12] is based on the idea that a force on the current pathway is generated by application of magnetic field to a liquid or tissue-like media in which electric currents are carried. This force is known as the Lorentz force. When the applied magnetic field is alternating, the generated force on the internal currents is also alternating. This time-varying force in the medium generates acoustic wave fronts which propagate away from the current site. These vibrations are detected at the object surface using ultrasonic transducers.

In MAET [13-19] technique the combination of electrical impedance tomography and ultrasound imaging is performed. In the existence of static magnetic field an ultrasonic pulse is propagated into the object, then an electrical current is induced and the surface voltage is measured through the electrodes. The electrical impedance of the object is reconstructed by using this voltage. Furthermore, an electrical pulse is propagated into the object and an ultrasound signal is detected with ultrasonic transducers.

In MAT-MI [20-23] technique, two magnetic fields are applied to the body: one static and one time-varying. Due to time-varying magnetic fields, eddy currents are generated. Eddy currents and applied static field result in vibrations (due to Lorentz force), and ultrasound waves are emitted. Ultrasonic transducers on the body surface are used as receivers. High resolution images of the conductivity distribution are reconstructed using this approach. The only difference between MAT-MI and MAT is that MAT uses electrical excitation under only static magnetic field. Since these approaches use ultrasound as the detected signal, the resultant conductivity images have high resolution.

AIMS OF THE INVENTION

The aims of the invention are:

-   -   Obtaining images of conductivity and permittivity distributions         of human body.     -   Obtaining multi-frequency images.     -   Obtaining a high resolution imaging system coupling static         magnetic field and ultrasound,     -   Obtaining a non-invasive imaging system for patients,     -   Obtaining an imaging system that operates within safety limits.

BRIEF DESCRIPTION OF THE INVENTION

In this invention, an approach is proposed to image the electrical impedance (conductivity and permittivity) properties of biological tissues. This approach is based on electrical current induction using ultrasound together with an applied static magnetic field. Acoustic vibrations are generated via piezoelectric transducers located on the surface of a biological body. A linear phased array piezoelectric transducer is used to form pressure distribution in human body/tissue. In the existence of a static magnetic field, the resultant (velocity) current density is sensed by a receiver coil encircling the tissue or placed near the tissue and used for reconstructing the conductivity and permittivity distribution.

DESCRIPTION OF THE FIGURES

The Figures and the related descriptions for the invention, multi frequency electrical impedance imaging system using Lorentz fields, described with this document are given below:

FIG. 1: Geometry of the proposed approach

FIG. 2: Proposed Electrical Impedance Imaging system

FIG. 3: Detailed block diagram of the system for the proposed electrical impedance imaging method

FIG. 4: X-coil configuration

FIG. 5: Y-coil configuration

FIG. 6: A single inhomogeneity (square domain of conductivity 0.8221 S/m) located at the center of the body (at a distance of 2.5 cm from the transducers) (The transducer is placed on the top side of the conductive body.)

FIG. 7: A single inhomogeneity (square domain of conductivity 0.8221 S/m) located at the center of the body (at a distance of 2.5 cm from the transducers) (The transducer is placed on the left side of the conductive body.

FIG. 8: Five identical inhomogeneities (each square domain of conductivity 0.8221 S/m) located symmetrically in the imaging domain. One is at the center of the body; others are at 1 cm distance from the body surfaces (The transducer is placed on the top side of the conductive body.)

FIG. 9: Five identical inhomogeneities (each square domain of conductivity 0.8221 S/m) located symmetrically in the imaging domain. One is at the center of the body; others are at 1 cm distance from the body surfaces (The transducer is placed on the left side of the conductive body.)

FIG. 10: Applied voltage (V) to each piezoelectric crystal.

FIG. 11: The reconstructed image of Model 1 when the SNR is 40 dB (a) and 80 dB (b). Data is acquired using two transducer positions.

FIG. 12: The reconstructed image of Model 2 when the SNR is 40 dB (a) and 80 dB (b). Data is acquired using two transducer positions.

DESCRIPTION OF THE COMPONENTS/PARTS OF THE INVENTION

The parts and components in the figures of the invention, described in this document are numbered for better understanding. The numerals referred to in the following description correspond to the following:

-   -   1—Ultrasonic transducer (single element and linear phased array)     -   2—Object to be imaged     -   3—Receiver coil (x-coil and y-coil)     -   4—Block diagram of the system for the proposed electrical         impedance imaging method     -   5—Control unit of the imaging system     -   6—Signal Generator     -   7—Power Amplifier     -   8—Static Magnetic Field Generator     -   9—(Pre-)Amplifier+Lock-in Amplifier     -   10—Filter     -   11—(Post-)Amplifier     -   12—Data Acquisition Card (DAC)     -   13—Memory     -   14-Analog to Digital Converter (ADC)     -   15—Current path     -   16—Inhomogeneity (Tumor)

DETAILED DESCRIPTION OF THE INVENTION

The steps of operation for this invention are described in this document basically comprising;

-   -   Applying a uniform static magnetic flux density with the static         magnetic field generator (8),     -   Exciting the ultrasonic transducer (1) sinusoidally at its         resonance frequency, f_(r), with signal generator (6) and power         amplifier (7),     -   Steering the linear phased array transducer (1) with different         angles, (A pressure wave is generated inside the object (2) and         (velocity) current density is induced due to pressure and static         magnetic field inside the object (2))     -   Measuring the magnetic fields of the induced current by receiver         coils (3),     -   Amplify the measured signal by pre-amplifier and Lock-in         Amplifier (9),     -   Filtering the amplified signal by filter (10),     -   Amplifying the filtered signal by post-amplifier (11),     -   Transmitting the filtered signal to the DAC (12),     -   Converting the analog signal to the digital signal by ADC (14),     -   Controlling the digital signal with control unit (5),     -   Reconstructing the conductivity and permittivity images with a         computer program.

The theoretical background of the proposed approach is given below in details.

The proposed imaging modality is a multiphysics problem. To develop a numerical model the electromagnetic and acoustics fields must be solved simultaneously (FIG. 1). First the basic field equations governing the behavior of time-varying electromagnetic and acoustic fields are given. Secondly, the general formulation of the partial differential equations for the scalar and magnetic vector potentials are presented in the electromagnetic part of this invention. In the acoustic part, the formulation of the partial differential equation for the acoustic pressure is presented. Thereafter, relation of the measurements to the existing (coupling) electromagnetic and acoustic waves is described.

1.1. BASIC ELECTROMAGNETIC FIELD EQUATIONS

The following set of Maxwell's equations governs the behavior of time-varying electromagnetic fields in a linear, non-magnetic, isotropic conductive body [24]:

$\begin{matrix} {{\nabla{\times \overset{\rightarrow}{E}}} = {- \frac{\partial\overset{\rightarrow}{B}}{\partial t}}} & \left( {E.\mspace{14mu} 1} \right) \\ {{\nabla{\times \overset{\rightarrow}{B}}} = {{\mu_{0}\overset{\rightarrow}{J}} + {\mu_{0}\frac{\partial\overset{\rightarrow}{D}}{\partial t}}}} & \left( {E.\mspace{14mu} 2} \right) \end{matrix}$ ∇·{right arrow over (D)}=ρ  (E.3)

∇·{right arrow over (B)}=0  (E.4)

For the solution of these fields we need the continuity condition

$\begin{matrix} {{\nabla{\cdot \overset{\rightarrow}{J}}} = {- \frac{\partial\rho}{\partial t}}} & \left( {E.\mspace{14mu} 5} \right) \end{matrix}$

and the constitutive relations:

{right arrow over (D)}=ε{right arrow over (E)}  (E.6)

{right arrow over (J)}=σ{right arrow over (E)}+σ({right arrow over (v)}×{right arrow over (B)})  (E.7)

where {right arrow over (v)} is the velocity of the conductor and {right arrow over (v)}×{right arrow over (B)} is the Lorentz field. The second term on the right hand side of this equation is known as the velocity current density.

Since the divergence of {right arrow over (B)} is zero, it is possible to introduce a magnetic vector potential {right arrow over (A)} as

{right arrow over (B)}=∇×{right arrow over (A)}  (E.8)

Consequently, {right arrow over (E)} can be expressed in terms of the magnetic vector potential {right arrow over (A)} and gradient of a scalar potential φ as

$\begin{matrix} {\overset{\rightarrow}{E} = {{- {\nabla\phi}} - \frac{\partial\overset{\rightarrow}{A}}{\partial t}}} & \left( {E.\mspace{14mu} 9} \right) \end{matrix}$

In three-dimensional (3D) problems, the electric and magnetic fields are usually calculated using a {right arrow over (A)}−φ formulation which results in two coupled equations in terms of {right arrow over (A)} and φ. To obtain the first equation, Equation (E.2) is rewritten in terms of the magnetic vector potential {right arrow over (A)} as follows:

$\begin{matrix} {{\nabla{\times {\mu_{0}^{- 1}\left( {\nabla{\times \overset{\rightarrow}{A}}} \right)}}} = {\overset{\rightarrow}{J} + \frac{\partial\overset{\rightarrow}{D}}{\partial t}}} & \left( {E.\mspace{14mu} 10} \right) \end{matrix}$

The terms {right arrow over (J)} and {right arrow over (D)} on right hand side can be expressed using equations (E.6) and (E.7) yielding:

$\begin{matrix} {{\nabla{\times \mu_{0}^{- 1}\left( {\nabla{\times \overset{\rightarrow}{A}}} \right)}} = {{\sigma\left( {\overset{\rightarrow}{E} + {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}}} \right)} + {\frac{\partial}{\partial t}\varepsilon \mspace{11mu} \overset{\rightarrow}{E}}}} & \left( {E.\mspace{14mu} 11} \right) \end{matrix}$

Reorganizing the right hand side we obtain,

$\begin{matrix} {{\nabla{\times \mu_{0}^{- 1}\left( {\nabla{\times \overset{\rightarrow}{A}}} \right)}} = {{\left( {\sigma + {\varepsilon \frac{\partial}{\partial t}}} \right)\overset{\rightarrow}{E}} + {\sigma\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right)}}} & \left( {E.\mspace{14mu} 12} \right) \end{matrix}$

By replacing {right arrow over (B)} and {right arrow over (E)} using the expressions in (E.8) and (E.9) we obtain,

$\begin{matrix} {{{\nabla{\times {\mu_{0}^{- 1}\left( {\nabla{\times \overset{\rightarrow}{A}}} \right)}}} + {\left( {\sigma + {\varepsilon \frac{\partial}{\partial t}}} \right)\left( {{\nabla\phi} + \frac{\partial\overset{\rightarrow}{A}}{\partial t}} \right)} - {\sigma\left( {\overset{\rightarrow}{v} \times {\nabla{\times \overset{\rightarrow}{A}}}} \right)}} = 0} & \left( {E.\mspace{14mu} 13} \right) \end{matrix}$

A second equation relating {right arrow over (A)} and φ can be obtained using the continuity equation (Equation (E.5)),

$\begin{matrix} {{\nabla{\cdot \overset{\rightarrow}{J}}} = {{\nabla{\cdot {\sigma\left( {\overset{\rightarrow}{E} + {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}}} \right)}}} = {{- \frac{\partial}{\partial t}}{\nabla{\cdot \left( {\varepsilon \mspace{11mu} \overset{\rightarrow}{E}} \right)}}}}} & \left( {E.\mspace{14mu} 14} \right) \\ {{{\nabla{\cdot {\sigma\left( {\overset{\rightarrow}{E} + {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}}} \right)}}} + {\frac{\partial}{\partial t}{\nabla{\cdot \left( {\varepsilon \mspace{11mu} \overset{\rightarrow}{E}} \right)}}}} = 0} & \left( {E.\mspace{14mu} 15} \right) \\ {\nabla{\cdot \left\lbrack {{{\left( {\sigma + {\frac{\partial}{\partial t}\varepsilon}} \right)\overset{\rightarrow}{E}} + {\sigma\left( {\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}} \right)}} = 0} \right.}} & \left( {E.\mspace{14mu} 16} \right) \end{matrix}$

Once again, using the expressions in equations (E.8) and (E.9), we obtain

$\begin{matrix} {{\nabla{\cdot \left\lbrack {{\left( {\sigma + {\frac{\partial}{\partial t}\varepsilon}} \right)\; \left( {{\nabla\phi} + \frac{\partial\overset{\rightarrow}{A}}{\partial t}} \right)} - {\sigma \left( {\overset{\rightarrow}{v} \times {\nabla{\times \overset{\rightarrow}{A}}}} \right)}} \right\rbrack}} = 0} & \left( {E.\mspace{14mu} 17} \right\rbrack \end{matrix}$

Equations (E.13) and (E.17) represent the general form of the system of equations that is used to calculate {right arrow over (A)} and φ for arbitrary excitations. For sinusoidal excitations (e^(jωt) is assumed), these two equations can be written using phasor notation as follows:

∇×μ₀ ⁻¹(∇×{right arrow over (A)})+(σ+jωε)(∇φ+jω{right arrow over (A)})−σ({right arrow over (v)}×∇×{right arrow over (A)})=0  (E.18)

∇[(σ+jωε)(∇φ+jω{right arrow over (A)})−σ({right arrow over (v)}×∇×{right arrow over (A)})]=0  (E.19)

In the proposed system, the conductive body (2) is source-free, i.e., the low-frequency biological sources and corresponding potentials are of no concern. The potentials {right arrow over (A)} and φ can be solved once the boundary conditions are set due to external sources. In the proposed approach, neither a time-varying magnetic field is applied using external coils nor a current or potential is applied using, for example, surface electrodes. The origin of the electric and magnetic potentials is the Lorentz fields and currents generated by the combination of a uniform static magnetic field {right arrow over (B)}₀ (say, in z direction) and a propagating acoustic field. The latter is generated by an ultrasound transducer (1) attached on the surface of the body (2) as shown in FIG. 1. Equations (E.18) and (E.19) together with appropriate boundary conditions can be used to calculate the electric field components for general conditions, i.e., general material properties and excitation frequencies.

1.2. BASIC ACOUSTIC FIELD EQUATIONS

To provide simplicity, a lossless, source-free medium is assumed and, initially, a one-dimensional (1-D) derivation is presented. In the static case, the body pressure is constant and denoted by p₀. The mass density of the body is assumed position dependent and represented by ρ₀(x). In the presence of a propagating pressure wave, the total pressure p_(T) is position and time-varying and is given as

p _(T)(x,t)=p ₀ +p(x,t)  (E.20)

In such a case, the total mass density can be expressed as

ρ_(T)(x,t)=ρ₀+ρ(x,t)  (E.21)

A propagating ultrasound also results in displacements in the small elements of the body and an associated ‘particle velocity’, {right arrow over (v)}(x,t). The particle displacement and its derivatives are small when |p/p₀|<<1 and |ρ/ρ₀|<<1. This results in a linearization procedure, and the ‘equation of motion’ is expressed as [25]:

−∂p/∂x=ρ ₀ ∂v/∂t  (E.22)

This shows that a particle accelerates in the opposite direction of the pressure gradient. The ‘continuity equation’ for mass conservation is written as

$\begin{matrix} {{- \frac{\partial\rho_{T}}{\partial t}} = \frac{\partial\left( {\rho_{T}v} \right)}{\partial x}} & \left( {E.\mspace{14mu} 23} \right) \end{matrix}$

which reduces to the following after linearization (under the above given conditions):

$\begin{matrix} {{- \frac{\partial\rho}{\partial t}} = {\rho_{0}\frac{\partial v}{\partial x}}} & \left( {E.\mspace{14mu} 24} \right) \end{matrix}$

To obtain a differential equation relating p and ρ, the particle velocity term in Equations (E.22) and (E.24) must be eliminated. This can be achieved by employing a ‘constitutive equation’ between them. If the change in mass density ρ is some function of changes in pressure p only, then a ‘linearized constitutive equation’ can be written as

ρ=[∂ρ/∂p] _(p=0) p=ρ ₀β₀ p  (E.25)

where β₀ is the compressibility (reciprocal of the bulk modulus or elastic constant) defined as

$\beta_{0} = {{- \frac{1}{V}}\frac{\partial V}{\partial p}}$

Three dimensional wave equation for pressure is derived by combining Equations (E.23), (E.24) and (E.25) as:

$\begin{matrix} {{\frac{1}{\rho_{0}c_{s}^{2}}\frac{\partial^{2}p}{\partial t^{2}}} = {\nabla{\cdot \left( {\frac{1}{\rho_{0}}{\nabla p}} \right)}}} & \left( {E.\mspace{14mu} 26} \right) \end{matrix}$

where c_(s) ²=(ρ₀β₀)⁻¹ represents the speed of pressure waves.

The equation of motion (Equation (E.22)) used in the preceding derivation is true as long as there are no other force terms. We may drop the terms related to gravitational force, whereas presence of current density {right arrow over (J)} and magnetic flux density {right arrow over (B)} in the body results in Lorentz force (per unit volume) {right arrow over (q)}={right arrow over (J)}×{right arrow over (B)} in addition to the mechanical forces in the body. Consequently, in the presence of a magnetic flux density and charged particles in the body (2), equation of motion can be modified as follows:

q _(x) ∂p/∂x=p ₀ ∂v/∂t  (E.27)

where q_(x) denotes the x-component of this interaction. Note that, movement of charged particles results in a current density {right arrow over (J)}=σ{right arrow over (E)}+σ({right arrow over (v)}×{right arrow over (B)}), in turn, current density influences the motion. In such a case, the wave equation should be modified taking into account the effects of Lorentz forces, yielding

$\begin{matrix} {{\frac{1}{\rho_{0}c_{s}^{2}}\frac{\partial^{2}p}{\partial t^{2}}} = {\nabla{\cdot \left\lbrack {\frac{1}{\rho_{0}}\left( {{\nabla p} - \overset{\rightarrow}{q}} \right)} \right\rbrack}}} & \left( {E.\mspace{14mu} 28} \right) \end{matrix}$

Since the medium is source-free, the pressure distribution is determined due to boundary conditions as dictated by an ultrasonic transducer (1) in contact with the medium. In this study, the boundary conditions are assumed as follows:

$\begin{matrix} {{\frac{1}{\rho_{0}}{\left( {{\nabla p} - \overset{\rightarrow}{q}} \right) \cdot \overset{\rightarrow}{n}}} = {a_{n}\mspace{14mu} {on}\mspace{14mu} S_{T}}} & \left( {E.\mspace{14mu} 29} \right) \end{matrix}$

where S_(T) denotes the body (2) surface which is in contact with the transducer, The term a_(n) in Equation (E.29) represents the local acceleration produced by the transducer (1) (derivation of this term is discussed in the next section). The remaining part of the surface is denoted by S. The second boundary condition is obtained using the continuity relation for acceleration:

${\frac{\partial{\overset{\rightarrow}{v}}_{1}}{\partial t} \cdot \overset{\rightarrow}{n}} = {\frac{\partial{\overset{\rightarrow}{v}}_{2}}{\partial t} \cdot \overset{\rightarrow}{n}}$

where {right arrow over (v)}₁ and {right arrow over (v)}₂ denote the particle velocities in region 1 and region 2 of an interface, respectively. This expression can also be written as

$\begin{matrix} {{\frac{1}{\rho_{01}}{\left( {{\nabla p} - \overset{\rightarrow}{q}} \right)_{1} \cdot \overset{\rightarrow}{n}}} = {\frac{1}{\rho_{02}}{\left( {{\nabla p} - \overset{\rightarrow}{q}} \right)_{2} \cdot \overset{\rightarrow}{n}}\mspace{14mu} {on}\mspace{14mu} S}} & \left( {E.\mspace{14mu} 30} \right) \end{matrix}$

1.3. PIEZOELECTRIC MEDIUM

Piezoelectric materials, which normally have neutral molecules, respond to an applied electric field by changing their mechanical dimensions.

Converse is also true; when a piezoelectric material is strained an electric field occurs due to small electric dipoles generated inside the material. This is due to their asymmetric atomic lattice. The field equations of piezoelectric materials couples the equations of elasticity and electricity by piezoelectric constitutive relations as explained below.

Force per unit area applied to a body is called stress and, in one-dimensional (1D) case, it is denoted by T. The fractional extension of the body is called strain and is denoted by S. For small stresses applied to a 1D system, the relation between stress and strain is given by the Hooke's law:

T=cS

where c=1/β₀ is the elastic constant of the material. In a piezoelectric material, however, the piezoelectric constitutive relation is as given below:

T=c ^(E) S−eE  (E.31)

The additional stress term on the right hand side is due to the presence of the electric field E. The parameter e is called the piezoelectric stress constant, and c^(E) is the elastic constant in the presence of constant or zero E field.

In the presence of an electric field E, the electrical displacement D depends on the strain as well as the electric field:

D=eS+ε ^(S) E  (E.32)

where ε^(s) is the permittivity with zero or constant strain.

If the material is not piezoelectric, the ‘equation of motion’, say in z-direction, is given by:

$\begin{matrix} {{\frac{\partial T}{\partial z} = {\rho_{0}\frac{\partial v}{\partial t}}}{or}} & \left( {E.\mspace{14mu} 33} \right) \\ {{c^{E}\frac{\partial^{2}u}{\partial z^{2}}} = {\rho_{0}\frac{\partial^{2}u}{\partial t^{2}}}} & \left( {E.\mspace{14mu} 34} \right) \end{matrix}$

which is another form of Equation (E.26) in terms of displacement u when the medium has uniform material properties.

The electrical behavior of the medium is governed by the following equation for the z component of the electrical displacement vector D_(z),

$\begin{matrix} {\frac{\partial D_{z}}{\partial z} = 0} & \left( {E.\mspace{14mu} 35} \right) \end{matrix}$

implying that D_(z) is constant and there are no free charges in the medium. The electric field can be written in terms of the derivative of a scalar potential as follows:

$E = {- \frac{\partial\phi}{\partial x}}$

Consequently, Equation (E.31) can be rewritten as:

$\begin{matrix} {{{- \varepsilon^{s}}\frac{\partial^{2}\phi}{\partial x^{2}}} = 0} & \left( {E.\mspace{14mu} 36} \right) \end{matrix}$

If the material is not piezoelectric, Equations (E.31) and (E.32) are isolated. In a piezoelectric material, however, the elasticity and electricity equations are coupled due to piezoelectric constitutive relations (E.31) and (E.32). In such a case, one obtains the following equations of piezoelectricity in 1D:

$\begin{matrix} {{{c^{E}\frac{\partial^{2}u}{\partial z^{2}}} + {e\frac{\partial^{2}\phi}{\partial z^{2}}}} = {\rho_{0}\frac{\partial^{2}u}{\partial t^{2}}}} & \left( {E.\mspace{14mu} 37} \right) \\ {{{e\frac{\partial^{2}u}{\partial z^{2}}} - {\varepsilon^{s}\frac{\partial^{2}\phi}{\partial x^{2}}}} = 0} & \left( {E.\mspace{14mu} 38} \right) \end{matrix}$

In a 3D problem, three equations are derived in the elasticity part for the three displacement components. Together with the electric field equation, four equations are obtained for four unknowns. Depending on the material properties, different forms of these equations can be found [26].

1.4 RELATION OF MEASUREMENTS TO THE CONDUCTIVITY DISTRIBUTION: MAGNETIC FIELD MEASUREMENTS (PROPOSED APPROACH)

Lead-Field analysis:

The measured data in the proposed approach are the voltages picked up from the receiver coil (3) (FIGS. 4 and 5). The relation between the receiver coil (3) voltage and magnetic flux density is given by the Faraday's law of induction:

$\begin{matrix} {{v_{ab}(t)} = {- {\int_{Coil}^{\;}{\frac{\partial\overset{\rightarrow}{B}}{\partial t}\  \cdot {\overset{\rightarrow}{S}}}}}} & \left( {E.\mspace{14mu} 39} \right) \end{matrix}$

where d{right arrow over (S)} is the differential surface element in the surface enclosed by the receiver coil (3). Since {right arrow over (B)}(t) is related to the conductivity and permittivity distributions in the body, the received voltage is a function of the conductivity and permittivity distributions. The above equation, however, does not show this relation explicitly. To obtain such a relation, in this study, an approach that is frequently used in formulating the forward problem of magnetoencephalography (MEG) is used. This approach is based on the reciprocity theorem [27]. In short, this theorem states that the location of the detector and source can be changed without affecting the detected signal amplitude. Using this theorem, the detected signal is expressed in terms of volume integral of the source (dipole) distribution. The sensitivity of the measurement to a specific dipole is determined by the scalar product of a lead-field vector (the electric field generated by a reciprocal unit current in the detector coil) with the selected dipole. Thus, once the lead field vector is solved for specific detector geometry, the detector voltage for an arbitrary dipole source can be easily calculated by integrating over the source domain.

Similar approach can be applied if the lead field vector for the proposed imaging modality can be identified. In the following sections, the lead field vector will be found for electromagnetic fields with 1) harmonic time dependence, and 2) general time dependence.

Harmonic Time Dependence:

A straightforward extension of this imaging modality is to excite the ultrasonic transducer (1) with its resonance frequency continuously. The pick-up voltage at the receiver coil (3) is then the steady state response (a single measurement) obtained for a specific transducer/receiver coil (3) configuration. To increase the number of measurements, the transducer and receiver coil (3) configurations should be changed. Though the measurement strategy is somehow difficult for continuous excitation, the analysis for the lead field vector analysis is relatively simple.

In the direct problem, {right arrow over (E)}₁ and {right arrow over (H)}₁ represent the electric field and magnetic field in the body (2) when currents are ultrasonically induced in the body (2). Assuming e^(jωt) time dependence and adopting the boldface phasor notation, the following equations are valid,

∇×{right arrow over (E)} ₁ =−jωμ ₀ {right arrow over (H)} ₁  (E.40)

∇×{right arrow over (H)} ₁ =σ{right arrow over (E)} ₁ +{right arrow over (J)} ₁  (E.41)

where {right arrow over (J)}₁=σ(v×{right arrow over (B)}₀) denotes the velocity current due to ultrasonic excitation under a static magnetic field.

In the reciprocal problem there are no current sources in the body (2). The electric field {right arrow over (E)}₂ and the magnetic field intensity {right arrow over (H)}₂ are generated inside the body (2) due to a reciprocal current density {right arrow over (J)}₂ (or {right arrow over (J)}_(R)) inside the receiver coil (3) of the direct problem. In the reciprocal problem the following equation must be satisfied:

∇×{right arrow over (E)} ₂ =−jωμ ₀ {right arrow over (H)} ₂  (E.42)

∇×{right arrow over (H)} ₂ =σ{right arrow over (E)} ₂ +{right arrow over (J)} ₂  (E.43)

Using Equations (E.40) and (E.43) one obtains the following equation:

H ₂ ·∇×{right arrow over (E)} ₁ −{right arrow over (E)} ₁ ·∇×{right arrow over (H)} ₂ =−jωμ ₀ {right arrow over (H)} ₁ ·{right arrow over (H)} ₂ −σ{right arrow over (E)} ₂ ·{right arrow over (E)} ₁ −{right arrow over (J)} ₂ ·{right arrow over (E)} ₁  (E.44)

Similarly, using Equations (E.41) and (E.42) we obtain

{right arrow over (H)} ₁ ·∇×{right arrow over (E)} ₂ −{right arrow over (E)} ₂ ·∇×{right arrow over (H)} ₁ =−jωμ ₀ {right arrow over (H)} ₁ ·{right arrow over (H)} ₂ −σ{right arrow over (E)} ₂ ·{right arrow over (E)} ₁ −{right arrow over (J)} ₁ ·{right arrow over (E)} ₂  (E.45)

Using the vector identity ∇×({right arrow over (A)}×{right arrow over (B)})={right arrow over (B)}·(∇×{right arrow over (A)})−{right arrow over (A)}·(∇×{right arrow over (B)}), the left hand sides of (E.44) and (E.45) are simplified and equations are put in the following form:

∇·({right arrow over (E)} ₁ ×{right arrow over (H)} ₂)=−jωμ ₀ {right arrow over (H)} ₁ ·{right arrow over (H)} ₂ −σ{right arrow over (E)} ₂ ·{right arrow over (E)} ₁ −{right arrow over (J)} ₂ ·{right arrow over (E)} ₁  (E.46)

∇·({right arrow over (E)} ₂ ×{right arrow over (H)} ₁ ______=−jωμ ₀ {right arrow over (H)} ₁ ·{right arrow over (H)} ₂ −σ{right arrow over (E)} ₂ ·{right arrow over (E)} ₁ −{right arrow over (J)} ₁·{right arrow over ({right arrow over (E)})}₂  (E.47)

By subtracting Equation (E.47) from Equation (E.46),

∇·({right arrow over (E)} ₁ ×{right arrow over (H)} ₂)−∇·({right arrow over (E)} ₂ ×{right arrow over (H)} ₁)={right arrow over (J)} ₁ ·{right arrow over (E)} ₂ −{right arrow over (J)} ₂ ·{right arrow over (E)} ₁   (E.48)

Taking the volume integral in all universe (in volume V_(∞) bounded by S_(∞)) we obtain

∫_(V) _(∞) ∇·({right arrow over (E)} ₁ ×{right arrow over (H)} ₂ −{right arrow over (E)} ₂ ×{right arrow over (H)} ₁)dV=∫ _(V) _(∞) ({right arrow over (J)} ₁ ·{right arrow over (E)} ₂₁ −{right arrow over (J)} ₂ ·{right arrow over (E)} ₁)dV  (E.49)

By applying the divergence theorem to the left hand side it becomes a surface integral,

∫_(V) _(∞) ∇·({right arrow over (E)} ₁ ×{right arrow over (H)} ₂ −{right arrow over (E)} ₂ ×{right arrow over (H)} ₁)dV=

{right arrow over (E)} ₁ ×{right arrow over (H)} ₂ −{right arrow over (E)} ₂ ×{right arrow over (H)} ₁ dS  (E.50)

Since the electric and magnetic field intensities vanish at infinity, we obtain the well-known reciprocity relation:

∫_(V) _(∞) ({right arrow over (J)} ₁ ·{right arrow over (E)} ₂ −{right arrow over (J)} ₂ ·{right arrow over (E)} ₁)dV  (E.51)

Since the current density {right arrow over (J)}₁ in the direct problem is nonzero in the conducting body volume (V_(body)) and the current density {right arrow over (J)}₂ in the reciprocal problem is nonzero in the receiver coil volume (V_(coil)), Equation (E.51) reduces to the following:

∫_(V) _(body) {right arrow over (J)} ₁ ·{right arrow over (E)} ₂ dV=∫ _(V) _(coil) {right arrow over (J)} ₂ ·{right arrow over (E)} ₁ dV  (E.52)

Recognizing that {right arrow over (J)}₂dV=I_(R)d{right arrow over (l)}, the right hand side becomes

∫_(V) _(body) {right arrow over (J)} ₁ ·{right arrow over (E)} ₂ dV=I _(R)∫_(V) _(coil) {right arrow over (E)} ₁ d{right arrow over (l)}  (E.53)

The integral on the right hand side is the pick-up voltage V_(ab) due to ultrasonically induced harmonic currents in the body (2). Consequently, V_(ab) is expressed as follows:

V _(ab)=∫_(V) _(body) {right arrow over (J)} ₁·({right arrow over (E)} _(R) /I _(R))dV  (E.54)

where {right arrow over (E)}₂ is replaced by {right arrow over (E)}_(R). The term in the parenthesis is the electric field in the reciprocal problem when unit current is applied to the receiver coil (3) and it is called as the lead field vector ({right arrow over (L)}_(M)) for the forward problem of the proposed imaging modality. The subscript Min the lead field vector shows that it is the lead field vector for the magnetic field measurements. Note that the lead-field vector itself is a function of body conductivity.

To show the relation between the pick-up voltage and conductivity in the medium, the term {right arrow over (J)}₁ is replaced by σ(v×{right arrow over (B)}₀) and the final form is obtained:

V _(ab)=∫_(V) _(body) σ(v×{right arrow over (B)} ₀)·{right arrow over (L)} _(M)(σ)dV  (E.55)

Note that the pick-up voltage is linearly proportional to the conductivity distribution and the weight of each conductive element is determined by the dot product of the Lorentz field and reciprocal field on that element.

General Time Dependence:

When general time dependence is assumed for the particle velocity V(t), the reciprocity relation given in Equation (E.51) does not hold. Instead, the following relation is valid [46]:

∫_(−∞) ^(∞) dt∫ _(V) _(body) dV{right arrow over (J)} ₁({right arrow over (r)},τ−t)·{right arrow over (E)} ₂({right arrow over (r)},t)=∫_(−∞) ^(∞) dt∫ _(V) _(coil) dV{right arrow over (J)} ₂({right arrow over (r)},t)·{right arrow over (E)} ₁({right arrow over (r)},τ−t)  (E.56)

In this expression, {right arrow over (E)}₁ is the electric field in the direct problem due to velocity current {right arrow over (J)}₁=σ({right arrow over (v)}(t)×{right arrow over (B)}) and {right arrow over (E)}₂ is the electric field in the reciprocal problem due to reciprocal current {right arrow over (J)}₂={right arrow over (J)}_(R) in the receiver coil. Here {right arrow over (r)} represents the three-dimensional position vector. For the receiver coil, we may write

∫_(V) _(coil) dV{right arrow over (J)} _(R)(t)=∫_(Coil) d{right arrow over (l)}I _(R)(t)

Then the right hand side of Equation (E.56) becomes,

∫_(−∞) ^(∞) dtI _(R)(t)∫_(Coil) E ₁(τ−t)·d{right arrow over (l)}=∫ _(−∞) ^(∞) dtI _(R)(t)V _(ab)(τ−t)=I _(R) *V _(ab)   (E.57)

For the left hand side of Equation (E.56), we recognize that the electric field {right arrow over (E)}_(R)({right arrow over (r)},t) in the reciprocal problem can be written as the product of a position dependent function and energizing reciprocal current I_(R)(t):

{right arrow over (E)} _(R)({right arrow over (r)},t)={right arrow over (E)} _(R) ⁰({right arrow over (r)})I _(R)(t)  (E.58)

where {right arrow over (E)}_(R) ⁰({right arrow over (r)}) is the reciprocal electric field normalized with unit current in the receiver coil (3). Since the body (2) is assumed resistive, Equation (E.58) should be valid. Consequently, we obtain

∫_(−∞) ^(∞) dt∫ _(V) _(body) dV{right arrow over (J)} ₁({right arrow over (r)},τ−t)·{right arrow over (E)} _(R)({right arrow over (r)},t)=∫_(−∞) ^(∞) dtI _(R)(t)∫_(V) _(body) dV{right arrow over (J)} ₁({right arrow over (r)},σ−t)·{right arrow over (E)} _(R) ⁰  (E.59)

Comparison of Equations (E.57) and (E.59) shows that

V _(ab)(τ−t)=∫_(V) _(body) dV{right arrow over (J)} ₁({right arrow over (r)},τ−t)·{right arrow over (E)} _(R) ⁰({right arrow over (r)})  (E.60)

A final form of this equation can be obtained by 1) change of variables for the time variation, 2) dropping the position vector {right arrow over (r)}, 3) expressing {right arrow over (J)}₁ in terms of particle velocity, and 4) setting {right arrow over (E)}_(R) ⁰={right arrow over (E)}_(R) ⁰(σ).

V _(ab)(t)=∫_(V) _(body) dVσ({right arrow over (v)}(t)×{right arrow over (B)} ₀)·{right arrow over (E)} _(R) ⁰(σ)  (E.61)

It is observed that for a given receiver coil (3) (and corresponding reciprocal field distribution), it is possible to acquire continuous data that gives information about the body conductivity along the propagation path of the particle velocity.

1.5. RECEIVER COIL DESIGN CONSIDERATIONS

When the ultrasonic transducer (1) position is fixed, the sensitivity in the measurements can be increased by optimizing the reciprocal field distribution. However, even for a fixed transducer (3) position ultrasonic beam steering approaches are applicable using an ultrasound array (1). Thus, in order to use a different receiver coil (3) for each beam direction, one should be able to acquire the best available data for any propagation direction. This can be achieved by using two coils for data acquisition, namely, x- and y-coils (3) which are sensitive to acoustic propagation in x- and y-directions, respectively. FIG. 4 and FIG. 5 shows two realizable coil (3) configurations encircling the body (1) for the development of x- and y-coils (3), respectively.

1.6. SIMULATION EXAMPLES

In this invention, the performance of the proposed imaging system (4) can be assessed by simulation studies. For simulation studies electrical and acoustic properties of homogeneous (2) and inhomogeneous (tumor) (16) bodies are defined. The conductive body (2) and tumor (16) are modeled as 5 cm×5 cm and 5 mm×5 mm in size, respectively. For conductive body (2) and tumor (16) electrical and acoustic properties of breast fat (2) and tumor (blood) (16) are represented. Electrical conductivities of breast fat (2) and tumor (blood) (16) are 0.0257 S/m and 0.8221 S/m, respectively at 1 MHz. The acoustic properties of breast fat (2) and tumor (16) are defined as: density of breast fat (2) is 980 kg/m³, the speed of sound in it is 1520 m/s; density of tumor (16) is 1040 kg/m³, the speed of sound in it is 1550 m/s.

To show the performance of the system, two models are prepared:

-   -   a) Model 1: A single inhomogeneity (16) (square domain of         conductivity 0.8221 S/m)) located at the center of the body (2)         (at a distance of 2.5 cm from the transducers (1)) (FIG. 6 and         FIG. 7)     -   b) Model 2: Five identical inhomogeneities (16) (each square         domain of conductivity 0.8221 S/m) located symmetrically in the         imaging domain (2). One is at the center of the body (2); others         are at 1 cm distance from the body (2) surfaces (FIG. 8 and FIG.         9)

A single transducer (1) (16-element linear phased array) is used for excitation. A sinusoidal voltage V is applied to the surface of each element of linear phased array (1) for one period of the excitation frequency f, i.e.,

V(t)=A sin(2πft) (t<1/f)  (E.62)

where the amplitude term A=1V and f=1 MHz (FIG. 10).

The Truncated Singular Value Decomposition (SVD) method is used to reconstruct the images of the conductivity distribution. Initially, a single transducer (1) (16-element linear phased array) is used for excitation as shown in FIG. 10. It is excited for seven steering angles (−22.5°, −15°, −7.5°, 0°, 7.5°, 15°, 22.5°). To assess the reconstruction performance in the case of noisy data Gaussian noise is added to the measurements. To equalize the resolution in both directions, data is acquired for a second transducer (1) position (FIG. 7). Images of the first model (Model 1) are reconstructed according to the Signal-to-Noise Ratio (SNR)s of 40 dB and 80 dB as shown in FIGS. 11 (a) and (b), respectively.

Similar procedure is implemented for Model 2. The reconstructed images according to SNRs of 40 dB and 80 dB are shown in FIGS. 12 (a) and (b), respectively.

2. REFERENCES

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1-19. (canceled)
 20. A high resolution, non-invasive imaging method to generate images of the conductivity and permittivity distribution of a specimen comprising the steps of: generating a static magnetic field with a static magnetic field generator; applying an acoustic signal wave to a specimen via a plurality of ultrasonic transducers; sensing the induced current due to the acoustic signal and the static magnetic field inside the specimen by a plurality of receiver coils; filtering, amplifying, transmitting; and converting an analog signal to a digital signal; and constructing a plurality of reconstructed images of conductivity and permittivity distribution of the specimen.
 21. The method according to claim 1, wherein the plurality of ultrasonic transducers is a single element or a linear phased array transducer.
 22. The method according to claim 1, wherein the plurality of receiver coils comprises coils which are sensitive to electrical current density, at least in the acoustic propagation direction and perpendicular to the acoustic propagation direction.
 23. The method according to claim 1, wherein the plurality of reconstructed images are single or multi-frequency conductivity or permittivity images.
 24. An apparatus to generate images of the conductivity and permittivity distribution of a specimen comprising: a static magnetic field generator configured to generate a uniform static magnetic field distribution; an ultrasonic transducer configured to generate acoustic waves in predetermined directions; and a plurality of receiver coils configured to sense the induced currents due to acoustic waves and static magnetic field inside the specimen; wherein the apparatus means for data acquisition for filtering, amplifying, transmitting, and converting analog signals to digital signals; and means for construction of a plurality of reconstructed images of conductivity and permittivity. 